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Thread: Internally Tangent Circle Problem Help

  1. #1
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    Internally Tangent Circle Problem Help

    Prove: If two circles are tangent internally at point P and the chords PA and (chord) PB of the larger circle intersect the smaller circle at points C and D respectively, then (chord) AB is parallel to (chord) CD.
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    Re: Internally Tangent Circle Problem Help

    What is your purpose in posting this? You show no attempt at all of your own to do the problem. If the two circles are "tangent internally" then the line perpendicular to the tangent line at P passes through the centers of both circles. That should make the problem easy.
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    Re: Internally Tangent Circle Problem Help

    I can draw the diagram I don't know how to prove it. Internally Tangent Circle Problem Help-img_0830.jpg
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    Re: Internally Tangent Circle Problem Help

    Quote Originally Posted by Ishven View Post
    Prove: If two circles are tangent internally at point P and the chords PA and (chord) PB of the larger circle intersect the smaller circle at points C and D respectively, then (chord) AB is parallel to (chord) CD.
    Quote Originally Posted by Ishven View Post
    I can draw the diagram I don't know how to prove it. Click image for larger version. 

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    http://mathhelpforum.com/newreply.ph...reply&p=921319


    In your you assumed that $A~\&~B$ are on opposites sides of $\overleftrightarrow {PO}$ where $O$ is the center of one of the circles.
    In the given they may well be on the same side. Look at my diagram.Internally Tangent Circle Problem Help-un4.gif
    Last edited by Plato; May 13th 2017 at 04:33 PM.
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    Re: Internally Tangent Circle Problem Help

    I believe your original statement is correct. I don't understand Plato's post.

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    Re: Internally Tangent Circle Problem Help

    Quote Originally Posted by johng View Post
    I don't understand Plato's post.

    The feeling is mutual. I have no idea how you are using the inscribed angle theorem. Do you mean the central angle theorem? In any case, I am sure the is a typo in your last sentence: $\Delta ABP~\&~\Delta ADP~???$
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    Re: Internally Tangent Circle Problem Help

    The theorem that I'm using is this one -- https://en.wikipedia.org/wiki/Inscribed_angle. Yes obviously, if you read the proof, there is a typo. It should read triangle ABP is similar to triangle CDP.
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    Re: Internally Tangent Circle Problem Help

    Quote Originally Posted by johng View Post
    The theorem that I'm using is this one -- https://en.wikipedia.org/wiki/Inscribed_angle. Yes obviously, if you read the proof, there is a typo. It should read triangle ABP is similar to triangle CDP.
    I of course read the proof. Aside from this diagram not being generic, I simply do not under how the inscribed angle theorem applies?Internally Tangent Circle Problem Help-un5.gif
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    Re: Internally Tangent Circle Problem Help

    As drawn, angle APB = angle CPD is acute. So $\angle AO_1B=2\,\angle APB=2\,\angle CPD=\angle CO_2D$. Similarly, if angle APB is obtuse. By the way, I realized this is not necessary to prove the similarity of triangles APB and CPD.

    Here's a diagram where the angle at P is obtuse. The proof is the "same".

    Last edited by johng; May 14th 2017 at 07:13 AM.
    Thanks from Plato
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